(I am assuming by "programming expert" you mean "a good software developer/engineer" by your usage of words like "build many constructive things out of it")
TL;DR version If you want to become a web/mobile application developer, you don't need to be very good at mathematics.
If you are interested in fields like Artificial Intelligence/Machine Learning/Statistical analysis, you will require good mathematical skills if not exceptional.
Long version Production-level applications are not just Algorithmics, there is also Software Design, Architecture and Engineering.
It all depends on the field you are more interested in: For web or database programming, they don't require you to be very good at mathematics.
For Artificial Intelligence, Machine Learning, Fuzzy Logic, you will require a good mathematical background about the basic concepts involved in them and they do require a good amount mathematical skills to excel in them.
If you consider programming languages: for most of the Declarative programming languages like SQL, CSS: you don't need to be very good at mathematics
For Imperative programming languages like C/C++/Java/Python: it depends on the kind of problem you are trying to solve, and requisite knowledge of mathematics is not directly related to the language but on the problem you are trying to solve. I would say the case is similar for Procedural programming, Object-oriented programming and Event-driven programming.
For Functional programming languages like Lisp/Haskell/Clojure/Erlang, a mathematical background is helpful in programming with these languages. I have not done a good amount of functional programming, hence I can't quite comment on the degree of mathematical expertise required for them.
To be called as an "expert", you will also need to be good at knowing whats wrong with your code (debugging) and check if your code is working as expected or not(testing).
Disclaimer: I have answered this question to the best of my knowledge and experience, in case of any wrong information conveyed in the answer, please notify by commenting :-)